Around 4:35 on "Using Cage Objects": You used the Subdivision Modifier "cranked up to 6" in Mode "Catmull Clark", then to avoid the smoothing, you increased the Mean-Crease in Edit Mode. I might be wrong, but can't you just use the "Simple" instead of Catmull-Clark Subdivision Method?
Nevermind, this is already answered in another Question, but I cannot delete my Post somehow.
